I have renters insurance and I accidentally left the water running in the bathroom, it backed up and and that flooded the wood floors eventually.
Is that going to be covered?
Legally who is responsible to repair those floor? Who comes and does the work?

Renters insurance is meant to cover the damage caused by negligence of the tenant. In this case if you need a speedy repair it is better to initiate steps yourself rather than putting the responsibility over the landlord. As you are covered by the insurance you can claim all the expenses through insurance.